Schroeren: Decoherent Histories of Spin Networks

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Abstract: The decoherent histories formalism, developed by Griffiths, Gell-Mann, and Hartle is a general framework in which to formulate a timeless, ‘generalised’ quantum theory and extract predictions from it. Recent advances in spin foam models allow for loop gravity to be cast in this framework. In this paper, I propose a decoherence functional for loop gravity and interpret existing results as showing that coarse grained histories follow quasiclassical trajectories in the appropriate limit.

2013-09-28 12:28:47: More on decoherence; focus on the histories of spin networks , that is spin foams. Main point: how to assign probabilities to such histories. Importance of decoherence in providing these quantities.

2013-09-28 12:31:46: general introduction to decoherence, in the context of closed quantum systems. Stress the difference with the classical situation in which a single history of the system is realized and is selected by the theory. It is difficult to assign probabilities to histories, in the quantum case, because of interference, mathematically the fact that we work with probabilities amplitudes for states and histories, and not probabilities themselves.

2013-09-28 12:32:52: In the Copenhagen interpretation, this interference is suppressed by measurement, via instantaneous “collapse” of the wave function to a given state.

2013-09-28 12:34:41: One motivation to use decoherence is that this Copenhagen story is questionable in many ways. One more motivation is that in the general relativistic context, no time variable can be used to describe when the collapse happens, and a covariant version of the suppression of superposition and interference should be looked for.

2013-09-28 12:36:07: Summary of idea of sum over histories as a way to describe in a covariant way the dynamics of a quantum system. Example of the [fine-grained] histories of a single particle and basic notion of coarse graining of such histories.

2013-09-28 12:38:33: More detailed definition of partition and coarse-graining of histories in this example. definition of class operator and decoherence functional for the histories. identification of the [decoherence] conditions that are necessary to be able to assign probabilities to the histories [basically amounting exactly to the suppression of interference between histories).

2013-09-28 12:41:06: More on the properties of the decoherence functional. Generalization of these concepts and definitions to the field theory [formulated in terms of path integrals] context and to GR [in particular in its Hamiltonian formulation].

2013-09-28 12:46:18: Move on to the covariant formulation of loop quantum gravity, that is to the spin foam formalism. Brief summary of the kinematical states of the theory [cylindrical functions and spin networks, as in lattice gauge theory]. Summary of the covariant definition of the dynamics: spin foams as histories of spin networks, defined by 2-complexes with the same type of coloring as spin network states.

2013-09-28 12:49:53: Summary of the dynamics: analog of path integral [sum-over-histories] is sum over all 2-complexes and sum over colorings, weighted by quantum amplitude.

2013-09-28 12:51:54: notion of coherent states as semiclassical states peaking on the classical phase space variables associated to a given graph [no superposition of graphs considered, here]

2013-09-28 12:52:02

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2013-09-28 12:58:13: Move on to the definition of the decoherence functional in this context. Fine-grained histories of spin networks are the individual spin foams, weighted by the “fundamental” spin foam amplitudes that the theory associated to them. Coarse graining is imposed on “bulk configurations”, that is on the spin foams being summed over. More precisely, one coarse grains all histories/spin foams that correspond to the same diff-invariant properties [identified in terms of suitable observables].

2013-09-28 13:00:31: The interesting and important question is to identify the quantities in terms of which one defines the coarse graining. One choice is to look at intrinsic and extrinsic curvature [as in the definition of coherent states], another is to look at volume operator [it has been considered in the loop quantum cosmology context].

2013-09-28 13:01:47: Comment by Dittrich: these observables are not diffeo-invariant, however. Indeed, an interesting issue, says the speaker.

2013-09-28 13:03:31: Can write down formaly the decoherence functional in terms of the sum over 2-complexes, which is however very difficult to control. Still, one can at least identify which properties such functional should satisfy.

2013-09-28 13:05:48: Application of general idea to the setting of “spin foam cosmology”, where results are available about obtaining semi-classicality in the spin foam setting. Lots of unsettled questions but interesting results to consider. Brief summary of framework and of approximations used.

2013-09-28 13:07:07: Conclusions: highlight two main open issues. One: dynamical – what physical processes lead to decoherence in this context. Two: conceptual – do spin foams resolve the problem of time?

2013-09-28 13:10:48: Butterfield: question on spin foam cosmology, about relaxing the various approximations and about obtaining more genaral anisotropic geometries. Vidotto: one can indeed generalize the class of states used, and use them to study anisotropic situations. A more difficult question is the dynamical one, i.e. relaxing the approximation to the dynamics that are used. But results are encouraging.

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One thought on “Schroeren: Decoherent Histories of Spin Networks

  1. Reposting a comment from 9/27/13

    “I would like to ask David Schroeren, why should one consider the Decoherent Histories (DH)approach in he context of Spin Networks (SN) when we know the very serious problems the general DH scheme suffers from?
    ( i.e. i) The fact that contradictory conclusions can be reached when considering different realms, and ii) the fact that there is no way in which the theory can point to any particular specific chose of realm which should be consider in any given situation ). Thus, it would seem, very problematic to say the least if the SN required such interpretational framework to work.

    Daniel Sudarsky”

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